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Kosterlitz-Thouless Transition of the XY model on rewired square lattices

Tasrief Surungan

Published Aug 14, 2016• 0 citations
OpenAlex

Abstract

The ferromagnetic XY model (O(2) symmetry) on decorated square lattices is \nstudied via Monte Carlo simulation to probe the role played by an increase in coordination \nnumber on the existing Kosterlitz-Thouless (KT) transition [1]. It is a unique transition between \nthe quasi-long range order (QLRO), also called the KT phase, and the paramagnetic phase at \nhigher temperature [2, 3]. The QLRO is the only type of order allowed by theMermin-Hohenberg \ntheorem for two dimensional (2D) spin model with continuous symmetry [4]. In diluted square \nlattices, where some links are depleted, it was found that the KT phase remains so long as the \nlinks are percolated. Here we probe whether the increase in average number of links preserves \nor destroys the KT phase. The decorated lattice is obtained by adding an extra link to each site \nof a square lattice, randomly connecting each site to one of its next-nearest neighbors [5]. The \naverage number of nearest links increases from two to three, which corresponds to an increase \nin dimensionality. The existence of KT phase is characterized by the correlation ratio, which \nwas also used in characterizing the critical properties of the 2D edge-cubic model [6]. It was \nfound that the KT phase shifted to higher temperature, which indicates that the KT phase on \ndecorated lattices is more robust against thermal fluctuation compared to that of the native \nsquare lattice. [3]. \nReferences

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Chinese Physics•Feb 1, 2004•1 citation
The ferromagnetic XY model (O(2) symmetry) on decorated square lattices is \nstudied via Monte Carlo simulation to probe the role played by an increase in coordination \nnumber on the existing Kosterlitz-Thouless (KT) transition [1]. It is a unique transition between \nthe quasi-long range order (QLRO), also called the KT phase, and the paramagnetic phase at \nhigher temperature [2, 3]. The QLRO is the only type of order allowed by theMermin-Hohenberg \ntheorem for two dimensional (2D) spin model with continuous symmetry [4]. In diluted square \nlattices, where some links are depleted, it was found that the KT phase remains so long as the \nlinks are percolated. Here we probe whether the increase in average number of links preserves \nor destroys the KT phase. The decorated lattice is obtained by adding an extra link to each site \nof a square lattice, randomly connecting each site to one of its next-nearest neighbors [5]. The \naverage number of nearest links increases from two to three, which corresponds to an increase \nin dimensionality. The existence of KT phase is characterized by the correlation ratio, which \nwas also used in characterizing the critical properties of the 2D edge-cubic model [6]. It was \nfound that the KT phase shifted to higher temperature, which indicates that the KT phase on \ndecorated lattices is more robust against thermal fluctuation compared to that of the native \nsquare lattice. [3]. \nReferences
Abstract The six-state clock model (SSCM) on rewired square lattice is studied using Monte Carlo simulation with Wang-Landau algorithm. This is a discrete counterpart of the well-known XY model, the native host of a unique topological phase transition called Kosterlitz-Tholess (KT) transition. The model has two KT transitions, i.e., at temperature T 1 and T 2, where T 1 < T 2. The first transition separates the lower temperature magnetic order and the quasi-long range order (QLRO) also known as KT phase; while the second transition separates the QLRO and the higher temperature paramagnetic phase. It has been established that the presence of KT phase is affected by the presence of randomness in the form of site and bond dilution. This intermediate phase is totally ruled out if bonds or sites of the lattice are no longer percolated. Here different type of randomness is probed, namely the rewired lattices, obtained by randomly adding one extra bond to each lattice site, and connect the site to one of its next-nearest neighbors. As a results, the average number of neighbors C increases. The increase of C affects the existence of KT phase. For each value of C, the KT temperatures, T 1 and T 2, were estimated from the plot of specific heats. Variation of KT temperatures for different values of C is observed, which is plotted with respect to each corresponding C to obtain the system phase diagram.
A two-dimensional triangular Ising lattice with the nearest-neighbor antiferromagnetic interaction J 1 and the next-nearest-neighbor ferromagnetic interaction - J 2 is studied by the Monte Carlo simulation. It is shown that the system with a finite J 2 undergoes the successive phase transition, paramagnetic→Kosterlitz-Thouless (KT)-like→ferrimagnetic phases, as temperature decreases. The sub-lattice switching phenomenon observed in our previous work is shown to be an intrinsic property inherent to the present system which has an XY character. The marginal exponent of the order-parameter correlation function at the KT-like transition point is seen to increase from 1/4 as the ratio J 2 / J 1 decreases, namely, it reveals a deviation from the universal behavior of a genuine XY model.
We study the two-dimensional XY\mathrm{XY}XY model with quenched random phases by Monte Carlo simulation and finite-size scaling analysis. We determine the phase diagram of the model and study its critical behavior as a function of disorder and temperature. If the strength of the randomness is less than a critical value, ${\ensuremath{\sigma}}_{c},$ the system has a Kosterlitz-Thouless (KT) phase transition from the paramagnetic phase to a state with quasi-long-range order. Our data suggest that the latter exists down to T=0T=0T=0 in contradiction with theories that predict the appearance of a low-temperature reentrant phase. At the critical disorder ${T}_{\mathrm{KT}}\ensuremath{\rightarrow}0$ and for $\ensuremath{\sigma}>{\ensuremath{\sigma}}_{c}$ there is no quasiordered phase. At zero temperature there is a phase transition between two different glassy states at ${\ensuremath{\sigma}}_{c}.$ The functional dependence of the correlation length on $\ensuremath{\sigma}$ suggests that this transition corresponds to the disorder-driven unbinding of vortex pairs.
We study the two-dimensional bond-diluted XYXYXY and six-state clock models by Monte Carlo simulation with cluster spin updates. Various concentrations of depleted bonds were simulated, in which we found a systematic decrease of the Kosterlitz-Thouless (KT) transition temperatures of both XYXYXY and six-state clock models as the concentration of dilution decreases. For the six-state clock model, a second KT transition at lower temperature was observed. The KT transition temperatures as well as the decay exponent $\ensuremath{\eta}$ for each concentration of dilution are estimated. It is observed that the quasi-long-range order disappears at the concentration of dilution very close to the percolation threshold. The decay exponent $\ensuremath{\eta}$ of the KT transitions calculated at each concentration indicates that the universality class belongs to the pure XYXYXY and clock models, analogous to the expectation of the Harris criterion for the irrelevance of randomness in the continuous phase transition of systems with nondiverging specific heat.
Convincing numerical evidence is obtained via extensive quantum Monte Carlo simulations on square lattices as large as 96\ifmmode\times\else\texttimes\fi{}96 that the spin-1/2 XY model undergoes a Kosterlitz-Thouless phase transition at kTc{\mathit{kT}}_{\mathit{c}}kTc​/J=0.350(4). Correlation length and in-plane susceptibility diverge at Tc{\mathit{T}}_{\mathit{c}}Tc​ precisely according to the form predicted by Kosterlitz and Thouless for the classical XY model. The specific heat increases very rapidly near Tc{\mathit{T}}_{\mathit{c}}Tc​ and exhibits a peak around kT/J=0.45.
We investigate phase transitions of the XY model on a two-layer square lattice which consists of a Villain plane ( J ) and a ferromagnetic plane ( I ), using Monte Carlo simulations and a histogram method. Depending on the values of interaction parameters ( I , J ), the system presents three phases: namely, a Kosterlitz–Thouless (KT) phase in which the two planes are critical for I predominant over J , a chiral phase in which the two planes have a chiral order for J predominant over I and a new phase in which only the Villain plane has a chiral order and the ferromagnetic plane is paramagnetic with a small value of chirality. We clarify the nature of phase transitions by using a finite size scaling method. We find three different kinds of transitions according to the values of ( I , J ): the KT transition, the Ising transition and an XY -Ising transition with ν = 0.849(3). It turns out that the Ising or XY -Ising transition is associated with the disappearance of the chiral order in the Villain plane.